Public Choice 98: 369–383, 1999. 369 c 1999 Kluwer Academic Publishers. Printed in the Netherlands. Nash equilibrium strategies in directional models of two-candidate spatial competition SAMUEL MERRILL, III 1 , BERNARD GROFMAN 2 & SCOTT L. FELD 3 1 Department of Mathematics and Computer Science, Wilkes University, Wilkes-Barre, PA 18766, U.S.A.; e-mail: smerrill@wilkes1.wilkes.edu; 2 School of Social Sciences, University of California, Irvine, Irvine, CA 92697, U.S.A.; 3 Department of Sociology, Louisiana State University, Baton Rouge, LA 70803, U.S.A. Accepted 31 January 1997 Abstract. The standard approach to two-party political competition in a multi-dimensional issue space models voters as voting for the alternative that is located closest to their own most preferred location. Another approach to understanding voter choice is based on preferred direction of change with respect to some specified neutral point (e.g., an origin or status quo point). For the two-dimensional Matthews directional model (Matthews, 1979), we provide geo- metric conditions in terms of the number of medians through the neutral point for there to be a Condorcet (undominated) direction. In this two-dimensional setting, the set of residual loca- tions for which no Condorcet directions exist is identical to the null dual set (Schofield, 1978) and to the heart (Schofield, 1993). In two dimensions, for most locations of the origin/status quo point, a Condorcet direction exists and points toward the yolk, a geometric construct first identified by McKelvey (1986). We also provide some simulation results on the size of the null dual set in two dimensions when the underlying distribution of points is non-symmetric. 1. Introduction There is a vast literature on spatial models of social choice. We may divide substantive applications into three subareas: (1) models of committee voting, (2) models of candidate competition, and (3) models of coalition formation. A number of the mathematical results in these literatures are equivalent or very similar, e.g., the search for the core of a spatial voting game is essentially equivalent to the search for an equilibrium location of candidates in a two- party competition and is closely related to the search for a stable coalition structure. We are indebted to Norman Schofield for helpful comments and to Dorothy Gormick and Chau Tran for library assistance. The paper was written while the first author was a Visiting Scholar in the Department of Biostatistics at the University of Washington. An earlier version of this paper was presented at the meeting of the Public Choice Society, 12–14 April 1996, in Houston, TX.